Re: Structures - GROUPS

"{Z, +, 0}" is the group of all integers with "+" as operation and "0" as the additive identity. {10, 15} "generate" the subgroup of all integers we can get by starting with 10, 15, -10, and -15 and adding them any number of times. If we call the number of times we add 10 n and the number of times we add 15 m, we get 10n+ 15m as a formula for all such numbers. In particular, we can write this as 5(2n+ 3m) showing that all numbers in this set are multiples of 5. What about the other way? If we can show that all multiples of 5 as in this set, we are done.
Suppose 10n+ 15m= 5k for some integer k. We can immediately divide by 5 to get 2n+ 3m= k.

Start by looking at 2n+ 3m= 1. It is obviously true that m= 1, n= -1 is a solution. Multipying by k gives m= k, n= -k such that 2n+ 3m= 2(-k)+ 3(k)= k. That is, given any integer k, there exist integers n and m such that 2n+ 3m= k and so that 10n+ 15m= 5k.

The subgroup generated by 10 and 15 is the subgroup of all multiples of 5.